#10340275

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#54823	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,5)(3,8,10,4)(6,13,15,9)(11,12,14,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,47)(45,46);(1,3)(2,8)(4,5)(6,14)(7,10)(9,12)(11,15)(13,16)(34,37)(39,41)(40,45)(47,48);(2,5)(3,10)(9,13)(11,14)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,35)(34,39)(36,44)(37,41)(38,43)(40,48)(42,46)(45,47);(1,4)(2,3)(5,10)(6,12)(7,8)(9,11)(13,14)(15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,34,38,37)(35,39,43,41)(36,40,46,45)(42,47,44,48);(1,5,7,2)(3,4,10,8)(6,13,15,9)(11,12,14,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,36,38,46)(34,40,37,45)(35,42,43,44)(39,47,41,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,16)(10,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,37,38,34)(35,41,43,39)(36,45,46,40)(42,48,44,47)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10247471);